Why Are AM, GM, and HM Important in Math?
Relation between AM, GM and HM can be derived with the basic knowledge of progressions or Mathematical sequences. An array or collection of objects in a specified pattern in Mathematics is called a Mathematical Sequence. A sequence is also referred to as a progression. The three most popular types of sequences are Arithmetic sequence, geometric sequence and harmonic sequence. An arithmetic sequence is a pattern of numbers in which the difference between consecutive terms of the sequence remains constant throughout the sequence. A geometric progression is a sequence of numbers in which any two consecutive terms of the sequence have a common ratio. Harmonic progression is the sequence that forms an arithmetic sequence when the reciprocal of terms are taken in order.
AM (Arithmetic Mean), GM (Geometric Mean) and HM (Harmonic Mean) are the most commonly used measure of central tendency. In Mathematics, when we learn about sequences, we also come across the relation between AM, GM and HM, where AM stands for Arithmetic Mean, GM stands for Geometric Mean, and HM stands for Harmonic Mean. The mean for any set is referred to as the average of the set of values present in that set. It is used to calculate growth rate and risk factors in finance, to calculate the rate of cell growth by division in biology, and to solve linear transformations.
AM, GM, HM stands for Arithmetic mean, Geometric mean and Harmonic mean respectively.
AM or Arithmetic Mean is the mean or average of the set of numbers which is computed by adding all the terms in the set of numbers and dividing the sum by a total number of terms.
GM or Geometric Mean is the mean value or the central term in the set of numbers in geometric progression. The geometric mean of a geometric sequence with ‘n’ terms is computed as the nth root of the product of all the terms in sequence taken together.
HM or Harmonic mean is one of the types of determining the average. The harmonic mean is computed by dividing the number of values in the sequence by the sum of reciprocals of the terms in the sequence.
AM, GM, HM Formula
Consider a sequence of ‘n’ terms as {a1, a2, a3, a4 …………. An}.
Case 1: If the above sequence is in arithmetic progression, the mean of this sequence is calculated as Arithmetic Mean using the formula.
AM = (a1+a2+a3+a4+........+an)/n
Case 2: If the given sequence is a geometric progression, the geometric mean of all the terms in the sequence is calculated using the formula.
\[GM = \sqrt[n]{a_{1}\times a_{2}\times a_{3}\times a_{4}\times.......\times a_{n}}\]
Case 3: If the sequence is in harmonic progression, the harmonic mean is computed by using the formula.
\[HM = \frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\frac{1}{a_{3}}+\frac{1}{a_{4}}+...+\frac{1}{a_{n}}}\]
Relation Between AM, GM and HM:
Consider two numbers ‘a’ and ‘b’ such that a and b are greater than 0. Terms in the sequence are ‘a’ and ‘b’ and the number of terms in the sequence ‘n = 2’. If AM GM HM formula is used, AM GM HM can be found as follows.
\[AM = \frac{(a+b)}{2}\]
\[GM = \sqrt{a+b}\]
\[HM = \frac{2}{\frac{1}{a}+\frac{1}{b}} = \frac{2}{\frac{b+a}{ab}} = \frac{2ab}{a+b}\]
The above equation gives the relation between AM,GM and HM. The equation can also be written as
\[AM \times HM = GM^{2}\]
or
\[GM = \sqrt{AM }\times HM\]
AM GM HM in Statistics has a vital role to play in major calculations.
Arithmetic Mean being very simple and easy to compute gives one of the measures of the central tendency of a grouped or ungrouped set of data.
Geometric mean is used in computation of stock indexes. Also geometric mean is used to calculate the annual returns of the portfolio. Geometric mean is also used in studying biological processes such as cell division and bacterial growth etc.
Harmonic mean is used to determine the price earnings ratio and other average multiples in Finance. It is also used in the computation of Fibonacci sequence.
Example Problems:
1. If five times the geometric mean of two numbers ‘a’ and ‘b’ is equal to the arithmetic mean of those two numbers such that a > b > 0, then compute the value of a + ba - b.
Solution:
Arithmetic mean of the two numbers is calculated as
\[AM = \frac{(a+b)}{2}\]
Geometric mean of the two numbers a and b is
\[GM = \sqrt{ab}\]
It is given in the question that Arithmetic mean = 5 times the geometric mean
AM = 5 GM
\[\frac{a+b}{2} = 5\sqrt{ab}\]
a + b = 10 √ab
(a + b)2 = 100 ab
(a - b)2 = (a + b)2 - 4 ab
(a - b)2 = 100 ab - 4 ab
\[(a-b)^{2} = 96ab\]
a-b = \[\sqrt{96ab}\]
\[\frac{a+b}{a-b} = \sqrt{\left ( \frac{100ab}{96ab} \right )} = \sqrt{\left ( \frac{25}{24} \right )} = 1.021\]
2. Find the harmonic mean of two numbers a and b, if their arithmetic mean is 16 and geometric mean is 8 provided that a > b > 0. (Hint: Use relation between AM GM HM formula).
Solution:
Given: AM = 16 and GM = 8
The relation between AM GM HM is given as:
\[AM \times HM = GM^{2}\]
\[16 \times HM = 8^{2} \]
\[16 \times HM = 64 \]
HM = \[\frac{64}{16} \]= 4
To find the numbers:
Arithmetic mean is given as
\[AM = \frac{(a+b)}{2}\]
\[16 = \frac{(a+b)}{2}\]
a + b = 32
a = 32 - b
Geometric mean is given as
GM = √ab
8 = √ab
64 = ab
64 = (32 - b) b
64 = 32b - b2
b2 - 32 b + 64 = 0
Fun Facts
It is inferred through a number of calculations and has been proved by experts who use AM GM HM in Statistics that the value of AM is greater than that of GM and HM. The value of GM is greater than that of HM and lesser than that of AM. The value of HM is lesser than that of AM and GM.
\[AM = \frac{(a+b)}{2}\]
\[GM = \sqrt{a+b}\]
\[HM = \frac{2ab}{a+b}\]
If zero is one of the terms of a sequence, its geometric mean is zero and the harmonic mean is infinity.
FAQs on Relation Between AM, GM, and HM Explained
1. What are the definitions of Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM)?
In mathematics, AM, GM, and HM are three different types of averages or measures of central tendency. The core difference lies in how they are calculated:
- Arithmetic Mean (AM): This is the most common type of average. It is calculated by summing all the values in a dataset and dividing by the count of values. For two numbers a and b, AM = (a+b)/2.
- Geometric Mean (GM): This is used for datasets that represent multiplicative growth, like interest rates. It is the nth root of the product of n numbers. For two numbers a and b, GM = √(ab).
- Harmonic Mean (HM): This is used for averaging rates or ratios. It is the reciprocal of the arithmetic mean of the reciprocals of the values. For two numbers a and b, HM = 2ab/(a+b).
2. What is the main formula connecting AM, GM, and HM for two positive numbers?
For any two positive numbers, the primary relationship connecting the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) is that the square of the Geometric Mean is equal to the product of the Arithmetic Mean and the Harmonic Mean. The formula is expressed as: GM² = AM × HM. This provides a direct mathematical link between the three means.
3. Which mean is the greatest: AM, GM, or HM?
For any set of distinct positive numbers, the Arithmetic Mean is always the greatest, followed by the Geometric Mean, and the Harmonic Mean is the smallest. This relationship is known as the AM-GM-HM inequality and is written as: AM ≥ GM ≥ HM. The equality (AM = GM = HM) holds true only in the specific case where all the numbers in the dataset are identical.
4. Can you explain the relation between AM, GM, and HM using a simple example?
Certainly. Let's take two positive numbers, 4 and 16. We can calculate each mean to see the relationship:
- AM = (4 + 16) / 2 = 20 / 2 = 10
- GM = √(4 × 16) = √64 = 8
- HM = (2 × 4 × 16) / (4 + 16) = 128 / 20 = 6.4
As you can see, the inequality AM > GM > HM (10 > 8 > 6.4) holds true. We can also verify the formula GM² = AM × HM: 8² = 64, and 10 × 6.4 = 64. The results match.
5. How are AM, GM, and HM connected to Arithmetic, Geometric, and Harmonic Progressions?
The means are fundamentally linked to their corresponding progressions. If you take any three consecutive terms of a progression, the middle term is the corresponding mean of the first and third terms:
- If three numbers a, b, and c are in an Arithmetic Progression (AP), then b is the Arithmetic Mean of a and c (i.e., b = (a+c)/2).
- If a, b, and c are in a Geometric Progression (GP), then b is the Geometric Mean of a and c (i.e., b = √ac).
- If a, b, and c are in a Harmonic Progression (HP), then b is the Harmonic Mean of a and c (i.e., b = 2ac/(a+c)).
6. Why is it important to prove that AM ≥ GM?
Proving the AM ≥ GM inequality is a fundamental concept in algebra and calculus because it is used to find the minimum or maximum values of many functions without using derivatives. For any two non-negative numbers a and b, the proof starts from the fact that the square of any real number is non-negative: (√a - √b)² ≥ 0. Expanding this gives a + b - 2√ab ≥ 0, which rearranges to (a+b)/2 ≥ √ab. This demonstrates that the Arithmetic Mean is always greater than or equal to the Geometric Mean, a powerful tool in optimization problems.
7. In what kind of real-world problems would you choose one mean over the others?
The choice of mean depends on the nature of the data and the question being asked. A good way to understand their importance is through application:
- Use AM for data that is additive and not heavily skewed, such as calculating the average score of a student across multiple tests.
- Use GM for calculating the average growth rate of something that grows multiplicatively, like financial investments or a bacterial population over several time periods.
- Use HM when you need to average rates or ratios, especially when the numerator is constant. A classic example is calculating the average speed for a journey where you travel the same distance at different speeds.
